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Theorem sorpss 6530
Description: Express strict ordering under proper subsets, i.e. the notion of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Assertion
Ref Expression
sorpss  |-  ( [ C.]  Or  A  <->  A. x  e.  A  A. y  e.  A  ( x  C_  y  \/  y  C_  x )
)
Distinct variable group:    x, y, A

Proof of Theorem sorpss
StepHypRef Expression
1 porpss 6529 . . 3  |- [ C.]  Po  A
21biantrur 494 . 2  |-  ( A. x  e.  A  A. y  e.  A  (
x [ C.]  y  \/  x  =  y  \/  y [ C.]  x )  <->  ( [ C.]  Po  A  /\  A. x  e.  A  A. y  e.  A  ( x [ C.]  y  \/  x  =  y  \/  y [ C.]  x ) ) )
3 sspsstri 3451 . . . 4  |-  ( ( x  C_  y  \/  y  C_  x )  <->  ( x  C.  y  \/  x  =  y  \/  y  C.  x ) )
4 vex 2961 . . . . . 6  |-  y  e. 
_V
54brrpss 6528 . . . . 5  |-  ( x [
C.]  y  <->  x  C.  y )
6 biid 229 . . . . 5  |-  ( x  =  y  <->  x  =  y )
7 vex 2961 . . . . . 6  |-  x  e. 
_V
87brrpss 6528 . . . . 5  |-  ( y [
C.]  x  <->  y  C.  x )
95, 6, 83orbi123i 1144 . . . 4  |-  ( ( x [ C.]  y  \/  x  =  y  \/  y [ C.]  x )  <->  ( x  C.  y  \/  x  =  y  \/  y  C.  x ) )
103, 9bitr4i 245 . . 3  |-  ( ( x  C_  y  \/  y  C_  x )  <->  ( x [ C.]  y  \/  x  =  y  \/  y [ C.]  x ) )
11102ralbii 2733 . 2  |-  ( A. x  e.  A  A. y  e.  A  (
x  C_  y  \/  y  C_  x )  <->  A. x  e.  A  A. y  e.  A  ( x [ C.]  y  \/  x  =  y  \/  y [ C.]  x ) )
12 df-so 4507 . 2  |-  ( [ C.]  Or  A  <->  ( [ C.]  Po  A  /\  A. x  e.  A  A. y  e.  A  ( x [ C.]  y  \/  x  =  y  \/  y [ C.]  x )
) )
132, 11, 123bitr4ri 271 1  |-  ( [ C.]  Or  A  <->  A. x  e.  A  A. y  e.  A  ( x  C_  y  \/  y  C_  x )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    \/ wo 359    /\ wa 360    \/ w3o 936   A.wral 2707    C_ wss 3322    C. wpss 3323   class class class wbr 4215    Po wpo 4504    Or wor 4505   [ C.] crpss 6524
This theorem is referenced by:  sorpsscmpl  6536  enfin2i  8206  fin1a2lem13  8297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-rpss 6525
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